## Research Directions

My research interests lie in the interdisciplinary area of non-equilibrium statistical and nonlinear physics, from complex biological systems to driven granular media. Biological multicellular systems are an exciting example of stochastic non-equilibrium systems, which exhibit fascinating nonlinear phenomena. My objective is to identify the basic physical mechanisms which govern complex biological processes.

### Emergent collective behavior of multicellular biological systems

Glioblastoma multiforme is a highly invasive and malignant brain tumor. Glioma cells not only divide (proliferate) but are also motile; a cell on a substrate is able to migrate its own diameter in 5-10 minutes. Thus, cells detach from the primary brain tumor and actively move away to the extracellular matrix. Therefore, even if the tumor core is taken out by a surgery, many invasive cells remain intact. Here we try to investigate collective migration of glioma cells by using both discrete stochastic lattice approach and a continuum modeling.

### Population dynamics

We investigate very rare, but highly important events: the role of large fluctuations in stochastic population dynamics.

### Granular flows

Driven granular matter is another fascinating example of intrinsically non-equilibrium systems. Fluidized granular media (media composed of inelastically colliding macroscopic particles) exhibit a variety of symmetry-breaking instabilities and pattern-forming phenomena. Understanding these instabilities is necessary for the development of quantitative models of granular flow, which have various industrial applications. Our research focuses on instabilities in driven granular gases as well as on phase separation in a dense shear granular flow. We investigate these problems employing Navier-Stokes granular hydrodynamics and verifying the results in molecular dynamics (MD) simulations.

## Project Descriptions

#### Migration of adhesive and proliferative brain tumor cells: investigation of front propagation

Consider first a simple discrete model for diffusion and proliferation. Each lattice site can be empty or once occupied. At each time step, a particle is picked at random. Then it can either jump to a neighboring empty site, or proliferate there (a new particle is born). We can ask for is the continuum analog of this model? It was shown that for proliferation rates the propagating fronts in this discrete system can be described by the Fisher-Kolmogorov equation. Suppose now that cells also experience cell-cell adhesion. Here, there are there are two interesting regimes. For subcritical adhesion, there are propagating "pulled" fronts, similarly to those of Fisher-Kolmogorov equation. For supercritical adhesion, there is a nontrivial transient behavior, where density profile exhibits a secondary peak.

In addition to random motion, a cell can perform directed motion in response to a gradient of some chemicals; this phenomenon is called chemotaxis. Chemotaxis can be positive if a cell moves toward a higher concentration of chemoattractants; during negative chemotaxis cells move toward the lower concentrations of chemorepellents. We have recently analyzed the phenomenon of cell migration by deriving a continuum equation for cell density from the underlying microscopic lattice model taking into account both cell-cell adhesion and chemotaxis. The theoretical predictions obtained by solving the resulting system of reaction diffusion equations agree very well with the numerical results of the stochastic hybrid model. We have also shown that when cell chemotaxis is taken into account, the theoretical results agree with the experimental data.

Read more:

- N. Charteris and E. Khain, "Modeling chemotaxis of adhesive cells: stochastic lattice approach and continuum description", New Journal of Physics 16, 025002 (2014).
- E. Khain, M. Katakowski, N. Charteris, F. Jiang, and M. Chopp, "Migration of adhesive glioma cells: Front propagation and fingering", Physical Review E 86, 011904 (2012).
- E. Khain and L. M. Sander, "Generalized Cahn-Hilliard equation for biological applications", Phys. Rev. E 77, 051129 (2008)
- E. Khain, L.M. Sander, and C.M. Schneider-Mizell, "The role of cell-cell adhesion in wound healing", J. Stat. Phys. 128, 209 (2007) - Special Issue on Applications to Biology
- T. Callaghan, E. Khain, L.M. Sander, R.M. Ziff, "A stochastic model for wound healing", J. Stat. Phys. 122, 909 (2006)
- E. Khain, L.M. Sander, and A. Stein, "A model for glioma growth", Complexity 11, 53 (2005)

#### Clustering of brain tumor cells

It turns out that invasive cells (that detached from the primary tumor and migrated away) have a very low proliferation (division) rate compared to those on the tumor surface, the so-called proliferative cells. This dichotomy between migration and proliferation is sometimes called the ``go or grow" property: that is, invasive cells mostly migrate rather than divide. Therefore, they are almost invisible to standard radiation and chemotherapy treatments, which kill cells that proliferate. The crucial problem in GBM treatment is that invasive cells may eventually switch back to the proliferative phenotype. This switch may occur after a cell has migrated a large distance (up to several centimeters) from the original solid tumor; it gives rise to recurrent tumors. The mechanisms of the phenotypic switch are poorly understood.

An attractive scenario is to see the phenotypic switch as a collective phenomenon. We have proposed that the phenotypic switch is related to the observed clustering of invasive cells. Once such clusters are formed in the invasive region, cells on the surfaces of the clusters can become proliferative again, like the cells on a surface of a primary tumor, thus leading to formation of distant recurrent brain tumors.

To investigate the mechanisms of cell clustering on a substrate, we formulated a discrete stochastic model for cell migration. The model accounts for cells diffusion, proliferation and adhesion. We predicted that cells typically form clusters if the effective strength of cell-cell adhesion exceeds a certain threshold. Another possibility is that recurrent brain tumors can be triggered by a rare event - spontaneous clustering of invasive tumor cells. Once a sufficiently large cluster is formed due to a large fluctuation, cells on the surface of the cluster may become proliferative, triggering rapid tumor growth.

Read more:

- E. Khain, M. Khasin and L. M. Sander, "The importance of rare events: spontaneous clustering of glioma cells as a trigger for tumor recurrence", (in preparation).
- E. Khain, C. M. Schneider-Mizell, M. O. Nowicki, E. A. Chiocca, S. E. Lawler and L. M. Sander, "Pattern formation of glioma cells: Effects of adhesion", EPL (Europhysics Letters) 88, 28006 (2009)
- E. Khain, L.M. Sander, and C.M. Schneider-Mizell, "The role of cell-cell adhesion in wound healing", J. Stat. Phys. 128, 209 (2007) - Special Issue on Applications to Biology

#### The role of hypoxia (lack of oxygen) in migration of brain tumor cells

We investigated the role of hypoxia in migration of brain tumor cells. We performed two series of cell migration experiments. The first set of experiments was performed in a typical wound healing geometry: cells were placed on a substrate, and a scratch was done. In the second set of experiments, cell migration away from a tumor spheroid was investigated. Experiments show a controversy: cells under normal and hypoxic conditions have migrated the same distance in the "spheroid" experiment, while in the "scratch" experiment cells under normal conditions migrated much faster than under hypoxic conditions. To explain this paradox, we formulate a discrete stochastic model for cell dynamics. The theoretical model explains our experimental observations and suggests that hypoxia decreases both the motility of cells and the strength of cell-cell adhesion. The theoretical predictions were further verified in independent experiments.

Read more:

- E. Khain, M. Katakowski, S. Hopkins, A. Szalad, X.G. Zheng, F. Jiang, M. Chopp, "Collective behavior of brain tumor cells: The role of hypoxia", Physical Review E 83, 031920 (2011).

#### Fingering in invasive tumor growth

We studied the in vitro dynamics of the malignant brain tumor glioblastoma multiforme. The growing tumor consists of a dense proliferating zone and an outer less dense invasive region. Experiments with different types of cells show qualitatively different behavior: one cell line invades in a spherically symmetric manner, but another gives rise to branches. We formulate a model for this sort of growth using two coupled reaction-diffusion equations for the cell and nutrient concentrations. When the ratio of the nutrient and cell diffusion coefficients exceeds some critical value, the plane propagating front becomes unstable with respect to transversal perturbations. The instability threshold and the full phase-plane diagram in the parameter space are determined. The results are in a qualitative agreement with experimental findings for the two types of cells.

Read more:

- E. Khain and L.M. Sander, "Dynamics and pattern formation in invasive tumor growth", Phys. Rev. Lett. 96, 188103 (2006).
*The paper has been selected for the May 15, 2006 issue of Virtual Journal of Biological Physics Research, published by the American Physical Society and the American Institute of Physics*

#### Fluctuations and stability in front propagation

We investigate the effects of large fluctuations in population dynamics. One example is effect of fluctuations on front propagation in a bistable system. Propagating fronts arising from bistable reaction-diffusion equations are a purely deterministic effect. Stochastic reaction-diffusion processes also show front propagation which coincides with the deterministic effect in the limit of small fluctuations (usually, large populations). However, for larger fluctuations propagation can be affected. We give an example, based on the classic spruce budworm model, where the direction of wave propagation, i.e., the relative stability of two phases, can be reversed by fluctuations.

Read more:

- E. Khain and B. Meerson, "Velocity fluctuations of noisy reaction fronts propagating into a metastable state", Journal of Physics A: Mathematical and Theoretical 46, 125002 (2013).
- E. Khain, Y. T. Lin, L. M. Sander, "Fluctuations and stability in front propagation", EPL (Europhys. Lett.) 93, 28001 (2011).

#### Minimizing the population extinction risk by migration

Many populations in nature are fragmented: they consist of local populations occupying separate patches. A local population is prone to extinction due to the shot noise of birth and death processes. A migrating population from another patch can dramatically delay the extinction. What is the optimal migration rate that minimizes the extinction risk of the whole population? Here, we answer this question for a connected network of model habitat patches with different carrying capacities.

Read more:

- M. Khasin, B. Meerson, E. Khain, and L. M. Sander, "Minimizing the population extinction risk by migration", Physical Review Letters 109, 138104 (2012).

#### Fast migration and emergent population dynamics

We consider population dynamics on a network of patches, having the same local dynamics, with different population scales (carrying capacities). It is reasonable to assume that if the patches are coupled by very fast migration the whole system will look like an individual patch with a large effective carrying capacity. This is called a "well-mixed" system. We show that, in general, it is not true that the total population has the same dynamics as each local patch when the migration is fast. Different global dynamics can emerge, and usually must be figured out for each individual case. We give a general condition which must be satisfied for the total population to have the same dynamics as the constituent patches.

Read more:

- M. Khasin, E. Khain, and L. M. Sander, "Fast migration and emergent population dynamics", Physical Review Letters 109, 248102 (2012).

#### Instabilities and fluid-solid coexistence in dense shear granular flow

It is known that transport coefficients of hard sphere fluid diverge at the density of dense close packing. However, there is recent evidence that the coefficient of shear viscosity diverges at a lower density than other constitutive relations. This may result in a coexistence of "solid-like" and "fluid-like" layers in dense shear flow. The density in "solid-like" layers is higher than the density of viscosity divergence, therefore these layers are at rest or move as a whole.

Read more:

- E. Khain, "Dense granular Poiseuille flow", Mathematical Modelling of Natural Phenomena, Volume 6, 77 - 86 (2011).
- E. Khain, "Clustering and phase separation in dense shear granular flow", Proceedings for the Third International Symposium on Bifurcations and Instabilities in Fluid Dynamics, Nottingham, 2009, Journal of Physics: Conference Series 216, 012008 (2010)
- E. Khain, "Bistability and hysteresis in dense shear granular flow", EPL (Europhysics Letters) 87, 14001 (2009)
- E. Khain, "Hydrodynamics of fluid-solid coexistence in dense shear granular flow", Phys. Rev. E 75, 051310 (2007)
- E. Khain and B. Meerson, "Shear-induced crystallization of a dense rapid granular flow: Hydrodynamics beyond the melting point", Phys. Rev. E 73, 061301 (2006)

#### Phase separation in vibrated granular monolayer

We investigate the long-standing puzzle of phase separation in a granular monolayer vibrated from below. Although this system is three-dimensional, an interesting dynamics occurs mostly in the horizontal plane, perpendicularly to the direction of vibration. Experiments [Olafsen and Urbach, Phys. Rev. Lett. 81, 4369 (1998)] demonstrated that for high amplitude of vibration the system is in the gas-like phase, but when the amplitude becomes smaller than a certain threshold, a phase separation occurs: a solid-like dense condensate of particles forms in the center of the system, surrounded by particles in the gas-like phase. We theoretically explain the experimentally observed coexistence of dilute and dense phases, employing Navier-Stokes granular hydrodynamics. We show that the phase separation is associated with negative compressibility of granular gas.

Read more:

- E. Khain and I. S. Aranson, "Hydrodynamics of a vibrated granular monolayer", Physical Review E 84, 031308 (2011).

#### Instabilities in driven granular gases

The phase-separation instability occurs in a very simple setting: an ensemble of inelastic hard disks driven by a rapidly vibrating side wall in the absence of gravity. This instability is surprisingly similar to the phase-separating instability in the van der Waals gas. Another instability is the thermal granular convection that develops in the same prototypical system, but in the presence of gravity. Convection in a horizontal layer of "classical" fluid heated from below, known as the Rayleigh-Benard convection, is a famous example of pattern formation outside of equilibrium. Understanding the analogous instability in a granular fluid is important for the physics of granular matter. I also investigated the oscillatory instability, where the system is driven by two opposite "thermal" walls at zero gravity. When the inelasticity of particle collisions exceeds a critical value, the "static" clustering state in the middle of the system becomes unstable and develops oscillations. Hydrodynamic predictions have been verified in MD simulations.

Read more:

- E. Khain, "Resonant oscillations of a granular cluster", Complexity 13, 45 (2008)
- E. Khain, B. Meerson, and P.V. Sasorov, , "Phase diagram of van der Waals-like phase separation in a driven granular gas", Phys. Rev. E 70, 051310 (2004)
- E. Khain,
**"Hydrodynamics of "thermal" granular convection"**, Proceedings of the Nato Advanced Research Workshop on "Continuum Models and Discrete Systems" (CMDS10), eds. D.J. Bergman and E. Inan (Kluwer Academic Publishers, 2004), p.341 - E. Khain and B. Meerson, "Oscillatory instability in a driven granular gas", Europhys. Lett. 65, 193 (2004)
- E. Khain and B. Meerson, "Onset of thermal convection in a horizontal layer of granular gas", Phys. Rev. E 67, 021306 (2003)
- E. Khain and B. Meerson, "Symmetry-breaking instability in a prototypical driven granular gas", Phys. Rev. E. 66, 021306 (2002)